The impression I got (though I don't really have anything to back this up) is that there is no actual organization of 'warpriests,' but rather there is this new armour, specially crafted and blessed in the 6th Age to recognize and honour the most devout warriors of these gods. In that way, 'warpriest' would be more an honorific for a fighter who has shown tremendous accomplishment/leadership/courage/whatever in the name of god. Don't ask me what this means for Tuska. I don't think we see any NPCs wearing this beyond the Lumby Crater priest.
I think the inspiration and role for the Temple Knights is meant to be pretty transparent. They're the Knights Templar, the most elite faction of warriors engaged in holy war, an order of the church, unbeholden to the laws of man, their rites and organization secretive and occasionally subject to suspicion. They're not priests, but they do have holy vows and take Saradomin as their highest authority.
Interestingly, the Knights Templar were the world's first international bank and multinational corporation - which ultimately led to their power becoming a threat to the king. Would love to see RS take some inspiration here.
I don't recall references to Paladins outside of the Ardougne boys. Plague's End establishes that Paladins explicitly serve the king, rather than the church or the principality, under the principle that he rules with the divine mandate of Saradomin. In that way, they're more of an honour guard.
White Knights are a 4th Age Asgarnian organization (though I guess Temple Knights adopted that as part of their ladder at some point). While Temple Knights exist to eliminate threats at any cost, at the imperative of Saradomin, I think White Knights are more beholden to Saradominism - the principles, not the god or his mouthpiece. They're more transparent, and explicitly a civic organization.
31-Mar-2017 06:27:59